RUI: Representation theory and homological algebra over local rings

RUI：局部环上的表示论和同调代数

• 批准号: 0901427
• 负责人: Janet Striuli
• 金额: $ 10.02万
• 依托单位: Fairfield University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901427&HistoricalAwards=false
• 关键词: RUI; Representation; theory; homological; algebra

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI will concentrate her attention on characterizing local Noetherian rings in terms of the behavior of certain subcategories of modules, extending her research to methods from homological algebra and representation theory. The idea stems from a successful description of rings of finite Cohen-Macaulay type, which flourished in the mid 1980s. Results about finite-dimensional algebras are generalized to rings of higher dimension. Such a description was recently extended to rings with a finite Gorenstein type. The PI?s main goal is to understand which categories of modules have the right properties to provide a description of the ring when finiteness conditions are available. To do so, the PI proposes to investigate further the relationship between the properties of the category of totally reflexive modules and those of the base ring. For example, the PI proposes to understand the long standing open problem to characterize which properties of the ring force every totally reflexive module to be free. As totally reflexive modules come from infinite complexes, the investigation naturally includes methods from homological algebra, where the properties of modules are linearized by a possibly infinite approximation.The project is devoted to investigating problems in commutative algebra with the main goal being to understand the set of solutions of polynomial equations in many variables. In the effort to describe such a set of solutions, algebraists study the set of functions defined on the set of solution when a visual intuition is not available. As functions can be added and multiplied, the set of functions has the structure of a commutative ring, of which the most familiar example is the ring of polynomials. It is well established that understanding a ring is tantamount to understanding the category of its modules, which are generalization of vector spaces over fields.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI将专注于表征局部诺特环的某些子类别的模块的行为，将她的研究扩展到同调代数和表示论的方法。这个想法源于对有限科恩-麦考利型环的成功描述，该环在20世纪80年代中期蓬勃发展。将有限维代数的结果推广到高维环上。这样的描述最近被扩展到有限Gorenstein型环。私家侦探？的主要目标是了解哪些类别的模块有正确的属性，以提供一个描述环时，有限性条件是可用的。为此，PI建议进一步研究全自反模范畴的性质与基环的性质之间的关系。例如，PI建议理解长期存在的开放问题，以表征环的哪些属性迫使每个全自反模块都是自由的。由于全自反模来自于无穷复形，研究自然包括同调代数的方法，其中模的性质通过可能的无穷近似线性化。该项目致力于研究交换代数中的问题，主要目标是理解多变量多项式方程的解集。为了描述这样的一组解，代数学家研究了当视觉直觉不可用时定义在解集上的函数集。由于函数可以相加和相乘，函数集具有交换环的结构，其中最常见的例子是多项式环。 众所周知，理解环等同于理解其模的范畴，模是域上向量空间的推广。